MSc in Photonics PHOTONICSBCN Universitat Politècnica de Catalunya (UPC) Universitat Autònoma de Barcelona (UAB) Universitat de Barcelona (UB) Institut de Ciències Fotòniques (ICFO) http://www.photonicsbcn.eu Master in Photonics MASTER THESIS WORK CLASSICAL SIMULATION OF RESTRICTED QUANTUM COMPUTATIONS Mrityunjaya Nebhwani Supervised by Dr. Matty Hoban and Prof. Antonio Acín (ICFO) Presented on date 9th September 2013 Registered at Classical simulation of restricted quantum computations Mrityunjaya Nebhwani Quantum Information Theory Group, ICFO – Institut de Ci`enciesFot`oniques,Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain E-mail: [email protected] Abstract. We study restricted models of measurement-based quantum computation; and we investigate whether their output probability distributions can be sampled from efficiently on a classical computer. We find that even for non-adaptive models of MBQC, if this task were feasible then a major conjecture of computational complexity theory would be violated. Finally, we show that our results coincide with previous work that is based on the quantum circuit model. 1. Introduction 1.1. Historical Background In 1982, Richard Feynman [11] observed that certain quantum mechanical phenomena could not be simulated efficiently on a Turing machine. He was trying to simulate the interaction of n particles under the laws of quantum mechanics, but any general solution that he found exploited exponential resources at best. Feynman pointed out that nature can carry out this simulation using nothing more than n particles, and suggested that this type of computation might be more efficient on a computing device that takes advantage of quantum mechanical e↵ects, and that perhaps even computation in general would be more efficient on a so-called quantum computer. David Deutsch then took these ideas further in 1985 [10] by introducing the universal quantum computer as the quantum analogue to the universal Turing machine. He showed that just as the universal Turing machine can efficiently simulate any Turing machine, so too can the universal quantum computer efficiently simulate any quantum computer. Deutsch also showed that the universal quantum computer can carry out computations that the universal Turing machine cannot, rendering it more powerful. Examples of these computations are the exact modelling of any physical process and the generation of genuine random numbers. In addition, Deutsch endorsed that a theory of quantum complexity, analogous to classical complexity theory, deserved further investigation. Bernstein and Vazirani did precisely this in their 1993 paper [5] and defined BQP, the primary quantum complexity class. Informally, BQP is the class of problems that are efficiently solvable by a quantum computer, just as P and BPP contain the problems that are feasible for a Turing machine and for a probabilistic Turing machine, respectively. They also highlighted that BQP contains BPP. Up until then, the interest in quantum computing was principally theoretical, but everybody became fascinated by the field when, in 1994, Peter Shor presented his now-famous algorithm [22, 23]. Shor’s algorithm can efficiently find the prime factors of an integer, a problem that has no known efficient classical solution. Furthermore, a quantum computer capable of executing Shor’s algorithm would be able to break many current cryptographic systems in a fraction of the time it would take a classical computer. In 1996, Seth Lloyd [17] proved Feynman’s 1982 conjecture [11], that a quantum computer can simulate any local quantum mechanical system. Shor’s and Lloyd’s results proved quantum computing as a revolutionary field at the crossroad of physics, mathematics, and computer science. However, the requirements for building a quantum computer are physically challenging for current technology. Is it possible to construct simpler quantum devices that already do something that a classical computer cannot? Recent work has given concrete evidence from the field of computational complexity that this may be possible [2, 6]. We will review this work and show that it belongs to a Classical simulation of restricted quantum computations 2 broader picture of simple quantum computations that are hard to classically simulate. First we review quantum computing and then we look at computational complexity. 1.2. Qubits and Quantum Logic Gates A qubit, or quantum bit, is the quantum analogue to the classical bit. Just as a bit, a qubit can take one of two possible values: 0 or 1. The di↵erence between both is that while a bit must be either 0 or 1, the laws of quantum mechanics allow a qubit to be 0, 1, or a superposition of both. When the system is measured, the system takes just one of these states with some probability. The two states that a qubit may take are known as basis states, and are written as 0 and 1 . The state of a qubit is thus described as a linear combination of its two basis states: | i | i = ↵ 0 + β 1 . (1) | i | i | i ↵ and β are complex numbers; when the qubit is measured, ↵ 2 and β 2 each give the probability of obtaining its associated state. The state is normalised so| it| follows| that| ↵ 2 + β 2 = 1. A quantum logic gate is an operator| thati acts on a small number of qubits,| | and| | is represented by a unitary matrix. For example, the Hadamard gate acts on a single qubit and its matrix representation is H (2). This gate is very useful as it maps each basis state to a superposition of both states (2). 1 11 0 + 1 0 1 H = H 0 = + | i | i H 1 = | i| i (2) p2 1 1 | i | i⌘ p2 | i |i ⌘ p2 ✓ − ◆ The Pauli gates (Pauli-X, Pauli-Y , and Pauli-Z) also act on a single qubit (3). The Pauli-X gate is the quantum equivalent of the NOT gate as it maps 0 to 1 , and vice-versa. | i | i 01 0 i 10 X = Y = Z = (3) 10 i −0 0 1 ✓ ◆ ✓ ◆ ✓ − ◆ As we shall see in the following sections, it is sometimes very useful to switch between the Pauli-X basis and the Pauli-Z basis. This can be done simply using X = HZH and Z = HXH. ✓ ✓ There are three other single-qubit gates that we will utilise here: RX and RZ are the rotations on the Bloch sphere by the angle ✓ about the x-axis and the z-axis, respectively (4). There is another single-qubit gate of note, the ⇡/8 gate, denoted T (4). ✓ ✓ i✓/2 cos( ) i sin( ) e− 0 10 R✓ = 2 − 2 R✓ = T = (4) X i sin( ✓ ) cos( ✓ ) Z 0 ei✓/2 0 ei⇡/4 ✓− 2 2 ◆ ✓ ◆ ✓ ◆ The controlled-NOT, or controlled-X, and the controlled-Z gates act on two qubits, and are examples of controlled gates. A controlled gate acts on a control qubit and a target qubit, performing some operation on the target qubit if and only if the control qubit is in the state 1 . For the controlled- NOT gate this operation is X, while for the controlled-Z gate this operation is|Zi (5). We can switch between CNOT and CZ by adding two Hadamard gates on the target qubit line: one before the controlled gate and one after. 1000 100 0 0100 010 0 CNOT = CZ = (5) 000011 0001 01 B0010C B000 1C B C B − C @ A @ A 1.3. Quantum Circuits A quantum circuit is a model for quantum computation. The input to the circuit is a set of input qubits. A fixed number of quantum gates is then applied to the qubits in order to perform the computation. Finally, the qubits are either output to act as input to another circuit, or they are measured. An example of a quantum circuit is displayed in Figure 1. This circuit creates an entangled 2-qubit Bell state β , with the qubit states x, y 0, 1 . Each line represents one qubit and time traverses from | xyi 2 { } Classical simulation of restricted quantum computations 3 x H | i • β | xyi y | i Figure 1: Quantum circuit to create Bell states. left to right. First the Hadamard gate is applied to x and then the controlled-NOT gate is applied to both the qubits; x and y being the control and target| i qubits, respectively. A small set of| i quantum| i gates is universal for quantum computation if any unitary operation can be well-approximated by a quantum circuit containing only those gates (by some appropriate notion of well-approximated). Thus, only these gates are necessary to build circuits that can execute any quantum computation. An example set of universal quantum gates is the Hadamard gate, the controlled-NOT gate, and the ⇡/8 gate [19]. In this work, we will consider quantum circuits that may not be universal for approximately simulating any unitary, but may be restricted in some way whether in the number or type of gates applied in a circuit. To be more formal we need to describe how a quantum circuit is formed given some classical bit-string x 0, 1 ⇤ of length x . This input is the same input that you would give to a classical computer but2 with{ } a quantum circuit| | we prepare the x -qubit input x along with p = f( x ) all in | | | i | | the state 0 where f is some function f : N N. Also, given x , we need a description of which gates to apply| i to which qubits; this description! can be captured by| | a list L( x ) of entries (U, (Q),t) indicating at which time-step t to apply a unitary U to which set of qubits (Q| )where| L( x )depends on x . For example, the circuit in Figure 1 is generated by the list L( x )=((H, 1, 1),|(Controlled-| NOT| ,|(1, 2), 2)).